In an FRW cosmology expressed in conformal coordinates the classical 'mechanical' action is as shown in the first image. The Dirac equation in conformal FRW spacetime is shown in the second image. In both cases the effect of the scale factor is to make the mass appear as if dynamic. (The 'spin connection' term results in a pure phase adjustment to the wavefunction and is of no import. )
In this coordinate system the EM fields are completely unaffected by the expansion; the EM action for the fields and the field-current interaction can be written as if in Minkowski spacetime.
Of course there is no new physics in switching from the traditional RW coordinate system to a conformal system. (In case you are concerned, though in the conformal system there is no Cosmological red-shift of radiation, there is instead a progressive blue shift of matter. Observationally these are indistinguishable. To make the same point in RW coordinates is messier, since both the matter and EM actions depend on the scale-factor.)
Mass renormalization offsets an infinite mass due to EM self-energy with an appropriately chosen mechanical mass so as to leave a finite positive 'observed' mass. Presumably the latter is the subject of the mechanical action, and scales accordingly with cosmological expansion.
But the electromagnetic part is unaffected by expansion. So it seems that renormalization in conformal FRW spacetime must cancel a constant infinity, and add a finite part that scales with expansion.
Note this would be a problem even without renormalization. I.E. even if the electromagnetic contribution to the mass was finite and non-zero, because the total mass would not then scale proportionally with expansion.
I presume that when the Higgs mechanism is adopted it is understood that the mechanical part of the mass is set to zero. Since the EM self-action will still be present the question above changes to one about the relative scaling with expansion of the Higgs mass and the electromagnetic self-energy of a charge. Specifically: the two contributions to the mass will not scale proportionally unless the Higgs field is conformally invariant.
Are the above inferences correct?
Do those that think about renormalization in curved spacetime have a work-around?
Hi, I have worked in conformal gravity and yes, the above inferences are correct: essentially it all comes down to state that the frame you choose is a conformal re-scaling, so all conformally invariant terms (massless Dirac equation, massless scalar equation and full electrodynamic equations) remain the same and all non-conformally invariant terms (all mass terms) scale accordingly. On the other hand, I never worked in renormalization groups so I cannot answer the second question.
Thank you Luca for your response.
Are you aware of any discomfort amongst those who are interested in such matters about the peculiar features demanded of renormalization in curved spacetime? Specifically, that it seems to make sense only if all contributions to the mass share the same dependence on a conformal scale factor?
... well, there is, but I think that this discomfort is not something specific to such a problem, rather it is a type of unease that more generally appears when we work with the methods of QFT in curved space-times.
- Badges
- Science topic
- Similar topics
- Physical Sciences
- Mechanics
- Classical Mechanics